Crossed Modules of Monoids II: Relative Crossed Modules

Abstract

This is the second part of a series of three strongly related papers in which three equivalent structures are studied:-internal categories in categories of monoids; defined in terms of pullbacks relative to a chosen class of spans -crossed modules of monoids relative to this class of spans -simplicial monoids of so-called Moore length 1 relative to this class of spans. The most important examples of monoids that are covered are small categories (treated as monoids in categories of spans) and bimonoids in symmetric monoidal categories (regarded as monoids in categories of comonoids). In this second part we define relative crossed modules of monoids and prove their equivalence with the relative categories of Part I.

A semi-abelian extension of a theorem by Takeuchi

“…Remark 5.6. The previous result can also be deduced from the recent results of Böhm (see Proposition 3.13 in [3], where a Hopf monoid in Vect is precisely a Hopf algebra).…”

Crossed Modules of Monoids I: Relative Categories

“…In these sequel papers we will apply this theory to categories of monoids in symmetric monoidal categories; that is, we consider relative categories of monoids. They will be shown to be equivalent to relative crossed modules of monoids (see [7]) and to relative simplicial monoids of Moore length 1 (in [8]).…”

Crossed Modules of Monoids III. Simplicial Monoids of Moore Length 1

“…Groups can be regarded as (distinguished) monoids in the cartesian monoidal category of sets. In our antecedent papers [3] and [4] we worked out the notion of crossed module of monoids in more general, not necessarily cartesian monoidal categories, relative to a chosen Communicated by George Janelidze. class of spans.…”

“…The main result can be found in Sect. 4 where we prove an equivalence between the category of relative categories in the category of monoids in C (cf. [3]) and the category of relative simplicial monoids in C whose Moore length is 1.…”

“…Throughout, we freely use definitions, notation and results from [3,4]. The composition of some morphisms A g B and B f C in an arbitrary category will be denoted by…”